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We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NP-complete problems the current upper and lower bounds lie exponentially ...

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. --- BQP is low for PP, i.e., PP BQP = PP. --- There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. --- There exists a relativized world where BQP does not have complete sets. --- There exists a relativized world where P = BQP but P 6= UP " coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

help us decide where and how to put our efforts into solving We show that MAEXP, the exponential time version of problems in complexity theory. It is still true that virtually the Merlin-Arthur class, does not have polynomial size cir- all of the theorems in computational complexity theory that cuits. This significantly improves the previous known result have reasonable relativizations do relativize (see [For94]). due to Kannan since we furthermore show that our result But we do have a small number of exceptions that arise does not relativize. This is the first separation result in com- from the area of interactive proofs. These results have preplexity theory that does not relativize. As a corollary to our viously always taken the form of collapses such as IP= separation result we also obtain that PEXP, the exponen- PSPACE [LFKN92, Sha92], MIP=NEXP [BFL91] and tial time version of PP is not in P=poly. PCP(O(1);O(logn))=NP [ALM+92]. In this paper we give the first reasonable nonrel-1

Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.

Is there a single-valued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to its second level. As the existence of such a function is known to be equivalent to the statement "every multivalued NP function has a single-valued NP refinement," our result provides the strongest evidence yet that multivalued NP functions cannot be refined. We prove our result via theorems of independent interest. We say that a set A is NPSV-selective (NPMV-selective) if there is a 2-ary partial function in NPSV (NPMV, respectively) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursion-theoretic notion of the semi-recursive sets and the extant complexity-the...

Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. In 1975, Simon proved that for unboundederror polynomial-time machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class, BPP path , and, more generally, we study the structural properties of BPP path . We prove that BPP path contains both NP BPP and P NP[log] , and that BPP path is contained in P \Sigma p 2 [log] , BPP NP , and PP. We conclude that, unless the polynomial hierarchy collapses, bounded-error threshold computation is strictly more powerful than bounded-error probabilistic computation. We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length. We show, for both BPP and BPP path , that if there is any database for which this formalization of security differs from the security given by oblivious database access, then P 6= PSPACE. It follows that if any set lacking small circuits can be securely accepted, then P 6= PSPACE.

We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman-Hartmanis isomorphism conjecture [BH77] holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP^A-complete sets are polynomial-time isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.

We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are P-separable.

The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools of computational complexity theory should come to bear on these important questions. Quantum Computing